Luquesio P. Jorge

The Barrier Principle for Minimal Submanifolds of Arbitrary Codimension

We establish a barrier principle for minimal submanifolds of a Riemannianmanifold of arbitrary codimension. We construct examples of barriers fortwo-dimensional minimal surfaces in ℝn, n ≥ 4, and apply these to deduceexistence as well as nonexistence theorems for Plateaus problem.

On the existence of complete bounded minimal surfaces in ℝn

Suppose there is a complete minimal surface M immersed in R%. By the Riemann-Koebe theorem the universal covering Ai~of M is either the complex plane or the unit disc. Since the coordinate functions of the immersion are harmonic and positive the first alternative is ruled out. Hence, in order to prove the theorem it suffices to show that a conformally flat metric .q on the unit disc D with bounded curvature cannot be realized by a minimal immersion into R~.. Let us suppose, by way of contradiction, that i : ( D , g ) ~ R~. is such an immersion.

Fundamental tone estimates for elliptic operators in divergence form and geometric applications

We establish a method for giving lower bounds for the fundamental tone of elliptic operators in divergence form in terms of the divergence of vector fields. We then apply this method to the Lr operator associated to immersed hypersurfaces with locally bounded (r + 1)-th mean curvature Hr + 1 of the space forms Nn+ 1(c) of constant sectional curvature c. As a corollary we give lower bounds for the extrinsic radius of closed hypersurfaces of Nn+ 1(c) with Hr + 1 > 0 in terms of the r-th and (r + 1)-th mean curvatures. Finally we observe that bounds for the Laplace eigenvalues essentially bound the eigenvalues of a self-adjoint elliptic differential operator in divergence form. This allows us to show that Cheegers constant gives a lower bounds for the first nonzero Lr-eigenvalue of a closed hypersurface of Nn+ 1(c).

The Spectrum of the Martin-Morales-Nadirashvili Minimal Surfaces Is Discrete

We show that the spectrum of a complete submanifold properly immersed into a ball of a Riemannian manifold is discrete, provided the norm of the mean curvature vector is sufficiently small. In particular, the spectrum of a complete minimal surface properly immersed into a ball of R is discrete. This gives a positive answer to a question of Yau [22].

On Properness of Minimal Surfaces with Bounded Curvature

We show that immersed minimal surfaces in the euclidean 3-space with bounded curvature and proper self intersections are proper. We also show that restricted to wide components the immersing map is always proper, regardless the map being proper or not. Prior to these results it was only known that injectively immersed minimal surfaces with bounded curvature were proper.

Minimal Surfaces and Their Gauss Maps

In this paper we will discuss some classical results in minimal surfaces theory, related to the Gauss map of such surfaces. In the last section we will comment on some work in progress and some open problems related to one of these results.

Spectrum Estimates and Applications to Geometry

In 1867, E. Beltrami (Ann Mat Pura Appl 1(2):329–366, 1867, [12]) introduced a second order elliptic operator on Riemannian manifolds, defined by \(\Delta ={\mathrm{{div}\,}}\circ {{\mathrm {grad}\,}}\), extending the Laplace operator on \(\mathbb {R}^{n}\), called the Laplace–Beltrami operator. The Laplace–Beltrami operator became one of the most important operators in Mathematics and Physics, playing a fundamental role in differential geometry, geometric analysis, partial differential equations, probability, potential theory, stochastic process, just to mention a few. It is in important in various differential equations that describe physical phenomena such as the diffusion equation for the heat and fluid flow, wave propagation, Laplace equation and minimal surfaces.